Counterexamples to finite generation at chromatic height two

1 · The essentials

The manuscript challenges a proposed rule about building certain objects in topology from a standard starting object. The rule concerns a setting described as chromatic height two. For every sufficiently large prime number, the manuscript constructs an object that meets the stated conditions but cannot be built from that starting object using the allowed operations finitely many times. Those operations include combining objects, shifting them, forming cones and taking retracts. Nevertheless, the constructed object passes two comparison tests involving different ways of localizing it. The manuscript presents this as a counterexample to the Hahn–Wilson conjecture at height two. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a construction kit with one standard component and a fixed list of permitted assembly moves. The manuscript describes an object that cannot be assembled in finitely many such moves, even though two pairs of inspection procedures give matching results for it. The image captures the contrast between finite construction and agreement under comparisons, not the mathematical details.

Why it matters

The claimed counterexamples separate two properties: being finitely constructible from the specified standard object and satisfying the stated localization comparisons. In this setting, those comparisons do not suffice to ensure finite construction. This is the precise limitation on the proposed generation principle highlighted by the manuscript.

2 · Going further

The precise claim is that, for every sufficiently large prime $p$ , there is a connective, $p$ -complete spectrum $X$ of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of $BP ⟨ 2 ⟩_{p}^{\land}$ . In the repository summary, this exclusion means that $X$ cannot be obtained from that generator through finitely many sums, shifts, cones and retracts. The qualification about the specified standard form is part of the statement.

The same spectrum is claimed to satisfy both $L_{2}^{f} X ≃ L_{2} X$ and $L_{T (2)} X ≃ L_{K (2)} X$ . Thus the failure of finite generation occurs even with both localization comparisons in place. The source identifies the result as a counterexample to the Hahn–Wilson conjecture at height two, but does not give the conjecture's full formulation or additional historical context.

The abstract describes the argument only at the level of its construction goal: produce a spectrum with the required hypotheses, establish its exclusion from the generated subcategory, and establish the two comparisons. It does not explain the construction or the mechanism that obstructs finite generation, so a more detailed account of the argument is not available from this source. Only the main manuscript is supplied; no companion manuscripts or relationships among them are described.

Useful notions

Spectra in topology
Connectivity and p-completion
Chromatic height and fp-type
Thick subcategories and finite construction
Localization of spectra

What was known before / what is new

Before

The source names the Hahn–Wilson conjecture as the proposal being challenged. It identifies the relevant finite-generation question using a specified standard form of completed $BP ⟨ 2 ⟩$ . It supplies no account of earlier results or the conjecture's history.

New

The manuscript claims a counterexample of exact fp-type two for every sufficiently large prime $p$ , with connectivity and $p$ -completeness imposed. Each example lies outside the ordinary thick subcategory generated by the specified standard form of $BP ⟨ 2 ⟩_{p}^{\land}$ . The same examples are claimed to satisfy both stated localization comparisons.

Limits and assumptions

The claim applies to sufficiently large primes; the source gives no explicit threshold and makes no claim for every prime. Its hypotheses are connectivity, p-completeness and exact fp-type two, and its conclusion concerns the ordinary thick subcategory generated by the specified standard form of completed BP⟨2⟩. The supplied abstract does not include the construction or argument details. The main result is not formalized in Lean and has not been formally verified. The source does not state what remains open at other heights or at primes outside the asserted range.

Mini glossary

Finite generation: Here, the ability to build an object from the specified generator using finitely many sums, shifts, cones and retracts.
Ordinary thick subcategory generated by an object: In this statement, the collection of objects obtainable from that object by the allowed finite construction operations.
Localization comparisons: The two asserted equivalences $L_{2}^{f} X ≃ L_{2} X$ and $L_{T (2)} X ≃ L_{K (2)} X$ for the constructed spectrum.
Counterexample: An object that meets a proposed rule's hypotheses but fails its conclusion.

Keywords : Chromatic height two · Finite generation · Hahn–Wilson conjecture

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed $BP ⟨ 2 ⟩$ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Counterexamples to the Hahn-Wilson conjecture at height two

26/09/2026

To be verified

We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of $BP ⟨ 2 ⟩_{p}^{\land}$ . The same spectrum satisfies both localization comparisons $L_{2}^{f} X ≃ L_{2} X$ and $L_{T (2)} X ≃ L_{K (2)} X$ .

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